[english] practice language analysis essay

diff --git a/methods/transformations.md b/methods/transformations.md
index 05f5c3b08cd964cdd7d2f1c49afdc993b61b07dc..153441bd9f0c3f9f72c42f0cf75421c7dbfad56a 100644 (file)
--- a/methods/transformations.md
+++ b/methods/transformations.md
@@ -1,3 +1,9 @@
+---
+geometry: margin=2cm
+columns: 2
+author: Andrew Lorimer
+---
+
 # Transformation
 
 **Order of operations:** DRT - Dilations, Reflections, Translations
@@ -9,13 +15,7 @@
 - $k$ - translation of $k$ units parallel to $y$-axis or from $x$-axis
 - $h$ - translation of $h$ units parallel to $x$-axis or from $y$-axis
 - for $(ax)^n$, dilation factor is $1 \over a$ parallel to $x$-axis or from $y$-axis
-
-## Translations
-
-For $y = f(x)$, these processes are equivalent:
-
-- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f(x)$
-- replacing $x$ with $x − h$ and $y$ with $y − k$ to obtain $y − k = f (x − h)$
+- when $0 < |a| < 1$, graph becomes closer to axis
 
 ## Dilations
 
@@ -29,6 +29,18 @@ For the graph of $y = f(x)$, there are two pairs of equivalent processes:
 
 For graph of $y={1 \over x}$, horizontal & vertical dilations are equivalent (symmetrical). If $y={a \over x}$, graph is contracted rather than dilated.
 
+## Reflections
+
+- Reflection **in** axis = reflection **over** axis = reflection **across** axis
+- Translations do not change
+
+## Translations
+
+For $y = f(x)$, these processes are equivalent:
+
+- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f(x)$
+- replacing $x$ with $x − h$ and $y$ with $y − k$ to obtain $y − k = f (x − h)$
+
 ## Transforming $f(x)$ to $y=Af[n(x+c)]+b$#
 
 Applies to exponential, log, trig, power, polynomial functions.  
@@ -54,19 +66,19 @@ Function $x^{\pm {p \over q}}$ is odd if $q$ is odd
 
 | $n$ is even: | $n$ is odd: |
 | ------------ | ----------- |
-|![](graphics/parabola.png){#id .class width=50%} | ![](graphics/cubic.png){#id .class width=50%} |
+|![](graphics/parabola.png){#id .class width=20%} | ![](graphics/cubic.png){#id .class width=20%} |
 
 ### $x^n$ where $n \in \mathbb{Z}^-$
 
 | $n$ is even: | $n$ is odd: |
 | ------------ | ----------- |
-|![](graphics/truncus.png){#id .class width=50%} | ![](graphics/hyperbola.png){#id .class width=50%} |
+|![](graphics/truncus.png){#id .class width=20%} | ![](graphics/hyperbola.png){#id .class width=20%} |
 
 ### $x^{1 \over n}$ where $n \in \mathbb{Z}^+$
 
 | $n$ is even: | $n$ is odd: |
 | ------------ | ----------- |
-|![](graphics/square-root-graph.png){#id .class width=50%} | ![](graphics/cube-root-graph.png){#id .class width=50%} |
+|![](graphics/square-root-graph.png){#id .class width=20%} | ![](graphics/cube-root-graph.png){#id .class width=20%} |
 
 
 ### $x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
@@ -82,8 +94,8 @@ If $n$ is odd, it is an odd function.
 
 $$x^{p \over q} = \sqrt[q]{x^p}$$
 
-- if $p \gt q$, the shape of $x^p$ is dominant
-- if $p \lt q$, the shape of $x^{1 \over q}$ is dominant
+- if $p > q$, the shape of $x^p$ is dominant
+- if $p < q$, the shape of $x^{1 \over q}$ is dominant
 - points $(0, 0)$ and $(1, 1)$ will always lie on graph
 - Domain is:  $\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}$
 
@@ -104,6 +116,17 @@ Closed circle - point not included
 
 Addition of linear piecewise graphs - add $y$-values at key points
 
-Product functions:  
+Product functions:
+
 - product will equal 0 if one of the functions is equal to 0
-- turning point on one function does not equate to turning point on product
\ No newline at end of file
+- turning point on one function does not equate to turning point on product
+
+## Matrix transformations
+
+Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$.
+
+## Composite functions
+
+$(f \circ g)(x)$ is defined iff $\operatorname{ran}(g) \subseteq \operatorname{dom}(f)$
+
+